The Destiny 2 launch appears to have run fairly smooth in spite of massive amounts of players joining the servers during the first day and a half. VG247 are reporting on a number of known issues currently affecting the game and chief among them is a bug that prevents players joining the Nightfall strike mid-game from qualifying for completion. This means you'll have to join the strike from the start for it to count.
There is also a glitch at Nessus that has to do with where your Ghost is placed if you happen to fall off a cliff. Other than that it mainly appears to be quality of life concerns and things that Bungie will be able to sort shortly.
We're currently playing the game with a mind to review it within the next couple of days.